Past exam of the advanced Quantum Condensed Matter Physics course of the University of Cambridge / 2016 / Paper 1 / 1 / b / Solution

For a fixed allowed band/subband pair, parabolic dispersions give $E_c(\mathbf{k})-E_v(\mathbf{k})=E_{j,n}^{\mathrm{edge}}+{\hslash }^2{k}_{\parallel }^2/(2{\mu }_j)$. If the photon energy exceeds the edge, the contour is a circle of radius $k_{\parallel }$ and the gradient magnitude is ${\hslash }^2{k}_{\parallel }/{\mu }_j$. ThusBelow threshold the contour is absent. With the conventional $d^{2}k/(2\pi {)}^2$ normalization, the two-dimensional joint density of states per area and independent channel isWith a locally energy-independent optical matrix element, each allowed pair therefore adds a constant step to the continuum dielectric function in the approximation of the supplied equation. Different hole families need not have equal step heights: reduced masses, spin multiplicities and Bloch-state matrix elements differ. Frequency-dependent overall optical prefactors, nonparabolicity, broadening and excitonic Coulomb enhancement can modify the literal step shape outside this approximation.